Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

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Homework Statement


Suppose A is m*n matrix b1 and b2 are m*1 vector and the systems Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

Homework Equations


The Attempt at a Solution



I think Ax = b1 + b2 should be consistent but i don't know how to prove..
 
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iamzzz said:

Homework Statement


Suppose A is m*n matrix b1 and b2 are m*1 vector and the systems Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

Homework Equations


The Attempt at a Solution



I think Ax = b1 + b2 should be consistent but i don't know how to prove..

No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.
 
Dick said:
No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.

let x1 be the solution of Ax1=b1 and x2 be the solution of Ax2=b2

So Ax1+Ax2=b1+b2=A(x1+x2)
so (x1+x2)could be x
prove ?
Is this correct ?
 
iamzzz said:
let x1 be the solution of Ax1=b1 and x2 be the solution of Ax2=b2

So Ax1+Ax2=b1+b2=A(x1+x2)
so (x1+x2)could be x
prove ?
Is this correct ?

I would say let x1 be ANY solution to Ax=b. There may be more than one. But why are you asking "Is this correct?". What part of it are you worried about?
 
Dick said:
I would say let x1 be ANY solution to Ax=b. There may be more than one. But why are you asking "Is this correct?". What part of it are you worried about?
I mean does that prove the problem ?

Anyway thanks for the help
 
iamzzz said:
I mean does that prove the problem ?

Anyway thanks for the help

I'm just saying I would feel better if you KNEW it solved the problem instead of having to ask. Yes, it's fine.
 
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